2015 tavasz

Bevezetés a kvantum-informatikába és kommunikációba 2014/2015 tavasz Kvantumkapuk, áramkörök… 2015. február 26.

A kvantummechanika posztulátumai (1) 1. Állapotleírás Zárt fizikai rendszer aktuális állapota egy olyan    állapotvektorral írható le, amely komplex együtthatókkal rendelkezik, egységnyi hosszú a H Hilbert-térben (egy komplex lineáris vektortérben,amelyben értelmezve van a belső szorzat).

2. A rendszer időbeli fejlődése A zárt rendszer időbeli fejlődése unitér transzformációval írható le, amely csak a kezdő és végállapottól függ.

A kvantummechanika posztulátumai (2) 3. A mérés Legyen X a mérés lehetséges eredményeinek a halmaza. Egy mérés a mérési operátorok halmazával adható meg:

Μ  Μ x , x  X , Μ x   Ha a megmérendő rendszer állapota  , akkor annak a valószínűsége, hogy a mérés az x eredményt adja:

PX |     M Tx M x  A mérés után a rendszer állapota az alábbi lesz |

 

Μx  px

A kvantummechanika posztulátumai (3) 4. Összetett rendszer Ha V és Y a két kvantumrendszerhez rendelt Hilbert-tér, akkor az ebből a két rendszerből álló összetett rendszerhez a W  V  YHilbert-tér rendelhető.

st 1

Postulate (state space)

• The actual state of any closed physical system can be described by means of a so called state vector v having complex coefficients and unit length in a Hilbert space V, i.e. a complex linear vector space (state space) equipped with inner product.

nd 2 Postulate

(evolution)

• The evolution of any closed physical system in time can be characterized by means of unitary transforms depending only on the starting and finishing time of the evolution. • 2nd Postulate can be interpreted as v’ (t2) = U(t1, t2)v(t2) and v’ V . • The above definition describes the evolution between discrete time instants, which is more suitable in context of quantum computing. Its original continuous-time form is known as Schrödinger equation

• Relationship between H and U

rd 3

Postulate (measurement)

• Any quantum measurement can be described bymeans of a set of measurement operators {Mm}, where m stands for the possible results of the measurement. The probability of measuring m if the system is in state v can be calculated as • and the system after measuring m gets in state

•

Because classical probability theory requires that

•

Completeness relation:

th 4

Postulate (composite systems)

• The state space of a composite physical system W can be determined using the tensor product of the individual systems W = V  Y . Furthermore having defined v V and y  Y then the joint state of the composite system is w = v  y.

Slides for Quantum Computing and Communications – An Engineering Approach

Chapter 2 Quantum Computing Basics Sándor Imre Ferenc Balázs

Elementary Quantum Gates "Excellent!" I cried, "Elementary" said he. Watson and Holmes, in "The Crooked Man", The Memoirs of Sherlock Holmes, Sir Arthur Conan Doyle

Pauli gates • Input: • Pauli X (bit-flip) gate:

• Pauli Z (phase-flip) gate:

Pauli gates • Pauli Y (???-flip) gate:

• Geometrical interpretation of Pauli X gate: rotation around axis x in the Bloch sphere

• Phase gate:

Hadamard gate

• Hadamard gate is Hermitian i.e. • furthermore: • It is worth memorising:

,

Hadamard gate • n-qbit Hadamard gate whit input

• n-qbit Hadamard gate with arbitrary classical input

Hadamard gate and the superposition principle

General Description of the Interferometer "An idea is always a generalization, and generalization is a property of thinking. To generalize means to think." Georg Hegel

Generalised interferometer

Copyright © 2005 John Wiley & Sons Ltd.

Abstract quantum circuit of the generalised interferometer


Analysis 1


0

Analysis 2


Analysis 3


Analysis 4


Analysis 5

Analysis 6 : idealistic scenario : fully random operation

Entanglement "Wonder is from surprise, and surprise stops with experience." Bishop Robert South

A surprising quantum state • Based on the 4th Postulate, decompose the following two-qubit state

  a 00  b 11 ? • No such decomposition exists! • Two types of quantum states – product – entangled

?

The CNOT gate


• Upper wire: control • Lower wire: data

The CNOT gate • Truth table

• Master equation

Matrix

The CNOT gate as classical copy machine • Provided the data input is initialized permanently with then the CNOT gate emits a copy of the control input on each output! • Let’s try to copy the following state!

• The input joint state is • Using the superposition principle the output becomes

which is nothing else then an entangled pair!

The SWAP gate 1


The SWAP gate 2


The SWAP gate 3


The SWAP gate 4


Bell states (EPR pairs) • Let us investigate the CNOT as an entanglement generator!


Bell states

• Bell states are orthogonal!!!

Generalised quantum entangler


Remarks • Only one of the entangled qbits is enough to entangle another qbit to the previous set of qbits. • Entanglement cannot be produced using only classical communication!

Decoherence – Entanglement with the environment • The first two postulates are valid only for closed systems therefore entanglement with the environment can be very dangerous! • In order to demonstrate it we use the well-known quantum interferometer and assume that the photon traveling trough the interferometer is entangled with a butterfly outside the laboratory (i.e. the environment). • The environment is represented by • The flying rule of butterfly

Generalised interferometer


Decoherence and the interferometer • Before butterfly’s action

• After butterfly’s action

• Now, the Hadamard gate comes

Decoherence and the interferometer

• Attention!

and

• Assuming

is real

are not necessarily orthogonal

Decoherence and the interferometer

=

• If then the entanglement disappears. • However, if then the operation becomes fully random instead of being deterministic while the observer becomes miss leaded.

Schrödinger’s cat "When I hear about Schrödinger’s cat, I reach for my gun." Stephen Hawking

Source: http://www.cbs.dtu.dk/staff/dave/roanoke/schrodcat01.gif

2015 tavasz

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